First slide

Theory of equations

Question

If $α, β$ are the roots of ${ax}^{2} + bx + c = 0$ and $α + β, α^{2} + β^{2}, α^{3} + β^{3}$ are in G.P., when $∆ = b^{2} - 4 ac$ , then

Moderate

A

$∆ \neq 0$
B

$b ∆ = 0$
C

$cb \neq 0$
D

$c ∆ = 0$

Solution

${(α^{2} + β^{2})}^{2} = (α + β) (α^{3} + β^{3})$
${(\frac{b^{2} - 2 ac}{a^{2}})}^{2} = (\frac{- b}{a}) (\frac{- b^{2} + 3 abc}{a^{3}})$
$\Rightarrow 4 a^{2} c^{2} = {acb}^{2} \Rightarrow ac (b^{2} - 4 ac) As a \neq 0 \Rightarrow c ∆ = 0$

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